On discrepancy, intrinsic Diophantine approximation, and spectral gaps

Gorodnik, Alexander; Nevo, Amos

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 127-174

In the present paper we establish bounds for the size of the spectral gap for group actions on homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $\left(\tau \right)$ of congruence subgroups of arithmetic lattices in algebraic groups which are forms of ${\mathrm{SL}}_2$, using estimates in intrinsic Diophantine approximation which follow from Heath-Brown’s analysis of rational points on $3$-dimensional quadratic surfaces.

Dans le présent article, nous établissons des bornes pour la taille de l’écart spectral pour les actions de groupe sur les espaces homogènes. Notre approche est basée sur l’estimation des normes des opérateurs de moyennage appropriés, et nous développons des techniques pour établir des bornes supérieures et inférieures pour de telles normes. Nous montrerons que ce problème analytique est étroitement lié au problème arithmétique de l’établissement de bornes sur la divergence de distribution pour les points rationnels sur les variétés de groupes algébriques. Comme application, nous montrons comment établir une borne effective pour la propriété $\left(\tau \right)$ des sous-groupes de congruence des treillis arithmétiques dans les groupes algébriques qui sont des formes de ${\mathrm{SL}}_2$, en utilisant des estimations dans l’approximation diophantienne intrinsèque qui découlent de l’analyse de Heath-Brown des points rationnels sur des variétés quadratiques de dimension $3$.

Classification : 22E46, 11F70, 11F72
Keywords: Semisimple group, discrepancy, spectral gap, Diophantine approximation, automorphic representation

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